Algebra 2 Made Easy Handbook Common Core Standards Edition

Transcription

1 Algebra Made Eas Handbook Common Core Standards Edition B: Mar Ann Case B. S. Mathematics, M. S. Education 0 Topical Review Book Compan, Inc. All rights reserved. P. O. Bo 38 Onsted, MI This document ma not, in whole or in part, be copied, photocopied, reproduced, translated, or reduced to an electronic medium or machine-readable form without prior consent in writing from Topical Review Book Corporation or its author.

2 Acknowledgments M sincere thanks to those who helped me put Algebra II Made Eas, Common Core Edition, together. The people who contributed include Kimberl Knisell, Director of Math and Science in the Hde Park, NY School District for her organizing epertise and to Jennifer Criser-Eighm, Director of Humanities in Hde Park, for proofreading the grammar and punctuation; to m daughter, Debra Rainha who teaches math at Andover (MA) High School and to her colleague, Stephanie Ragucci who teaches AP Statistics at Andover High. Her assistance was invaluable. Marin Malgieri contributed to the unit on Probabilit and Statistics, and additional proofreading was done b Laura Denkins. Graphics designer, Julieen Kane, at Topical Review Book Compan alwas does a great job getting the graphs and diagrams done correctl as well as putting m drafts into the necessar form to be published. Keith Williams, owner of Topical Review Book Compan, is alwas eas to work with and ver patient. I am honored to have m work published b the Little Green Book compan familiar to man for providing Regents Eamination stud materials since 93. Introduction This quick reference guide, or how to do it book It is not meant as a curriculum guide to the Common Core Standards, nor is it meant to be used as a tetbook. The Common Core Standards (CCS) involve new methods of teaching and learning and use various methods for solving mathematical problems. Implementing new teaching techniques and providing deeper understanding of the content of the course is the job of the classroom teacher. As the Algebra II CCS are implemented, it is m hope that this student friendl book will assist students in becoming college and career read, as this is one of the main goals of our educational process. Sincerel, MarAnn Case, B.S. Mathematics, M.S. Education

3 ALGEBRA MADE EASY Common Core Standards Edition Table of Contents UNIT : POLYNOMIALS, RATIONAL, AND RADICAL RELATIONSHIPS.... Real Numbers and Eponents.... Radicals....3 Imaginar and Comple Numbers...3. Factoring Polnomials.... Quadratic Equations Rational Epressions Solving Rational Equations and Inequalities Solving Radical Equations Sstems of Equations Practice and Solving Equations Parabola: Focus and Directri...7 UNIT : TRIGONOMETRY FUNCTIONS Arc Length Angles In Quadrants Unit Circle and Trig Graphs...8. Graphing The Trig Functions Graphing Inverse Trig Functions Reciprocal Trig Functions Graphs Trig Identities and Equations...03 UNIT 3: FUNCTIONS Functions Graphs of Functions Functions and Transformations Average Rate of Change Inverses of Functions Eponential Functions Logarithmic functions Graphing Eponential and Logarithmic Functions Polnomial Functions Comparing Functions Sequences and Series...5

4 ALGEBRA MADE EASY Common Core Standards Edition Table of Contents UNIT : PROBABILITY AND STATISTICS Statistics...0. Statistical Measures The Shape of a Distribution.... Inference Organizing Categorical Data Displaing Bivariate Data ( Variables) Probabilit Tpes of Probabilit General Probabilit Rules Independent and Conditional Events Probabilities Using Two-Wa Table and Venn Diagrams...99 CORRELATIONS TO CCSS...0 INDEX...0

5 Unit POLYNOMIALS, RATIONAL, AND RADICAL RELATIONSHIPS Perform arithmetic operations with comple numbers. Use comple numbers in polnomial identities and equations. Interpret the structure of epressions. Understand the relationship between zeros and factors of polnomials. Use polnomial identities to solve problems. Rewrite rational epressions. Understand solving equations as a process and eplain the reasoning. Solve equations and inequalities in one variable. Solve sstems of equations. Analze functions using different representations. Translate between the geometric description and the equation for a conic section. Etend the properties of eponents to rational eponents. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved.

6 . REAL NUMBERS AND EXPONENTS In past math courses, we have worked with the real numbers. The real numbers consist of all the numbers on the number line. In this course, we will also stud numbers in the set of comple numbers. The two main subsets of the reals are the rational numbers and the irrational numbers. The rational numbers include all those numbers that can be written in the form of a fraction where both the numerator and the denominator are integers. The irrational numbers include the square roots of imperfect squares, decimal numbers that do not repeat and do not terminate, Pi (π), and e which is the base of natural logarithms. (See page 3.) A comple number can be written in the form a + bi where a and b are real numbers, i is the imaginar unit and i = -. (See Chapter.3 Imaginar and Comple Numbers) EXPRESSIONS WITH EXPONENTS Note: All the rules that appl to numbers with eponents are also used for epressions containing variables. Bases and Eponents: An eponent indicates how man times to use its base as a factor. The base of the eponent is the number, variable, or parenthesis directl to the left of the eponent. If the epression to the left of the eponent is a parenthesis, the eponent is applied to the entire content of the parenthesis. Evaluating with an eponent is often referred to as raising a number to a power. Review of Properties of Eponents: + a a = a ( a ) = a a = a, a 0 a ( ab) = ab Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. ( ) a b a =, b 0 b a 0 =, a 0 a =, a 0 a

7 Eamples. Bases and Eponents 3 = = 8 3 = ( ) = 8 The base is 3. The negative sign is not directl to the left of the eponent, therefore it is not affected b the eponent. ( 3) = ( 3)( 3)( 3)( 3) = 8 Here the base is the contents of the parenthesis, 3. 5 = 5 = 5 Onl the is the base for the eponent,. (5) = (5)(5) = 5 (5) is directl to the left of the eponent,, so the entire parenthesis is the base and is multiplied b itself to evaluate the epression. The same rule applies if the base is a fraction or a polnomial. Eample The Real Number Sstem ( 3 ) = ( 3 ) ( 3 ) = 9 is different than = = ( + ) = ( + )( + ) = 3 + +, ( + ) is the base. Negative Eponents: A negative eponent directs us to use the reciprocal of the base raised to the indicated eponent. The negative sign has nothing to do with the positive or negative value of the base. Once we translate the base into its reciprocal, the negative sign on the eponent disappears. The positive eponent is then applied as usual. Eamples 3 3 = ( ) ( )( )( ) = = 8 3( ) = 3( 3 3 ) ( )( = ) = ( ) =( ) ( )( ) = = ( ) = 5 ( ) ( )( ) = = Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 3

8 .0 SOLVE SYSTEMS ALGEBRAICALLY Substitution: One equation is manipulated so that or is isolated, then the resulting representation of that variable is substituted in the second original equation. The remaining variable is solved for, then that answer is substituted in either original equation to find the second variable. = and = 3 Steps: ) Solve the first equation ( = ) for : = + ) Get the second equation ( = 3) ( + ) = 3 and substitute ( + ) for in it. + = 3 Solve for : = = 3) Go back to an original equation: = ) Substitute for : ( ) = 5) Solve for : + = ) Indicate both answers: = and = 7) Checking both answers = = 3 in both original equations ( ) = ( ) = 3 is the final step: + = + = 3 = 3 = 3 Note: This method is recommended ONLY when the coefficient of or is. Coefficients other than result in fractional substitutions which must be cleared or the rapidl become unmanageable. Algebra Made Eas Common Core Edition 0 Copright 0 Topical Review Book Compan Inc. All rights reserved.

9 .0 Addition: If two equations have the same variable with opposite coefficients, we can add the equations together and variable. Sometimes it is necessar to multipl one equation make equivalent equations that can be used in this method. Steps: ) Arrange both equations using algebraic methods so the variables are underneath each other in position. ) The goal is to eliminate one variable b adding the two equations together. Eamine the variables and their coefficients. Find the least common multiple of the coefficients of either both s or both s. 3) Multipl each equation b a positive or negative number so the coefficients of the variable chosen are equal and opposite in sign. ) Add the two equations together. One variable will disappear. 5) Solve for the variable that is visible. ) Choose one of the original equations and substitute the value for the known variable to find the other variable. 7) Check in both original equations. Eample Solve the following equations for and : (A) + = (B) 3 = 8 Steps: ) is a common multiple for and 3 so leave (A) as ( 3 = 8) ; = 5 is and multipl (B) b. Reasoning with Equations and Inequalities ) Add (A) and (B) in order to isolate : Equation (A): + = NEW Equation (B) from step : = 5 8 = 8 ; = 3) Insert the answer back into either (A) or (B): () + = ) Solve for : = 0 ; = 0 5) Check in both original equations. () + (0) = () 3(0) = 8 = 8 = 8 Note: Eperience will help ou decide which variable to work with in the addition method. Looking for variables which alread have opposite signs allows ou to avoid multipling b a negative number with its associated opportunities for error. Using small multiples is helpful, too. You wouldn t want to use a common multiple for and if ou could use the other variable and have a common multiple of and 5. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved.

10 .0 WORD PROBLEM SYSTEMS WITH VARIABLES As in all word problems, read and reread. Make sure our equations represent the phrases in the wording of the problem.. Identif each unknown quantit and represent each one with a different variable in a let statement. READ CAREFULLY - make the let statement accurate.. Translate the verbal sentences into two equations. 3. Solve as a sstem of equations. Usuall these problems are solved algebraicall but follow directions - ou might be directed to solve them graphicall.. Check the answers in the words of the problem. Word Problem Sstems: Man word problems can be set up using two variables instead of using just one. If ou choose to use that method, ou must then make two equations to solve. The problem will then be solved as a sstem of equations. Eample Together Evan and Denise have 8 books. If Denise has four more than Evan, how man books does each person have? Let = the number of books Evan has Let = the number of books Denise has Steps: ) Set up equation (A): + = 8 ) Set up equation (B): = + 3) Use substitution: + ( + ) = 8 + = 8 ) Solve for : = = + = 8 5) Substitute in original: + = 8 ) Solve for : = Answer: Evan has books and Denise has books. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved.

11 5 = + (, 33).0 SOLVE SYSTEMS BY GRAPHING Graphing Sstems of Linear Equations: Two or more equations are graphed on the same coordinate plane (grid). Graph EACH equation separatel, but put both on one coordinate graph. Be ACCURATE. Label each line as ou graph it. The point where the intersect is the solution set of the sstem of equations. Label the point of intersection on the graph. This point is the solution set. Check both the and values of the solution in both original equations. The and values of the point of intersection must satisf both equations. Eample Solve this sstem of equations graphicall and check: (A) = + and (B) = + (These are both alread in = m + b form.) Steps: (A) = + (B) = + ) Determine the slope: slope = / slope = / ) Determine the -intercept: (0, ) (0, ) 3) Graph, label, and find solution set: SS = {(, 3)} ) Check: (A) = + (B) = + 3 = + 3 = ( ) + 3 = 3 3 = + 3 = 3 5 = + 5 Reasoning with Equations and Inequalities 5 Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 3

12 Unit TRIGONOMETRY FUNCTIONS Etend the domain of trigonometric functions using the unit circle. Model periodic phenomena with trigonometric functions. Prove and appl trigonometric identities. Summarize, represent and interpret data on two categorical and quantitative variables. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 79

13 . ARC LENGTH To find the length of the arc subtended (cut off b) b a central angle of a circle use this formula: s = r θ where s represents the arc length, r is the radius, and θ is the central angle MEASURED IN RADIANS. If the measure of the central angle is given in degrees, it must be changed to radian measure before using the formula. Eamples Find the length of the arc intercepted b θ if the radius is 0 cm and θ =. Since θ is alread in radian form, we don t need to change it. s = r θ s = (0)() s = 0 cm Find θ in radians if the arc length is and the radius is. s = r θ = θ θ = 3 radians Find the length of the arc, to the nearest tenth, cut off b a central angle measuring 75 if the radius of the circle is 5 cm. Hint: Change degrees to radians. 75 π radians 75π 5π = = radians ( ) 5π s = ( 5) =. 5 s 5. cm Algebra Made Eas Common Core Edition 80 Copright 0 Topical Review Book Compan Inc. All rights reserved.

14 .3 Special Angles: On the unit circle, as the terminal side of θ, (in standard position), is rotated counterclockwise, the (, ) values of each point of intersection with the unit circle are the cosine and sine values of the angle formed. Commonl, the angles that are multiples of 30, 5, 0 and the quadrantal angles are used to sketch the graphs of the trig functions. θ in Radians cos θ sin θ tan θ (sin θ/cos θ) 0 * 0 0 π π π π * undefined 0 π π π π * 0 0 7π π π π * 0 undefined π π π * 0 0 * Indicates a quadrantal angle Algebra Made Eas Common Core Edition 88 Copright 0 Topical Review Book Compan Inc. All rights reserved.

15 .5 GRAPHING INVERSE TRIG FUNCTIONS Remember, a function must be one-to-one for it to have an inverse function. To find the inverse of a function, the (, ) coordinates of the points in the function are reversed and then the become the points on the inverse. If a point on f() is (, 7), the corresponding point on f (), the inverse, is (7, ). The graph of f () is the reflection of the graph of f() over the line =. Domain and Range of trig functions and their inverses. The domain of the function is the range of its inverse. The range of the function is the domain of its inverse. In the function, the domain shows the possible values of the angle. The range is the values of the trig function. In the inverse, the domain is the possible values of the trig function, and the angles values are the range. The domain of the sine, cosine, and tangent functions must be restricted. Eamples : : Function sin Inverse arc sin or sin Domain : : Range : : Trigonometric Functions f() = sin () g() = sin () π 0 π π π - π 0 π Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 99

16 Unit 3 FUNCTIONS Understand the concept of a function and use function notation. Interpret functions that arise in applications in terms of the contet. Analze functions using different representations. Write a function in different forms to eplain different properties of the function. Build a function that models a relationship between two quantities. Build a new function from eisting functions. Construct and compare linear, quadratic, and eponential models and solve problems. Interpret epressions for functions in terms of the situation the model. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 05

17 Practice: Find the domain and the range where possible. It is often helpful to use a graph to find the range. Determine if the equation is a function or not, and if it is a function, is it one-to-one? -5 Eamples = , = Domain: : Range: { : > 0} The graph shows the domain and range. The horizontal line test does not work so it is not one-to-one. The vertical line test works: 3 = is a function. - 7 Domain: : 3 Range: { : 0} It passes the vertical line test and the horizontal line test. This is a one-to-one function. Building and Interpreting Functions = Domain: No restrictions. The domain is the set of real numbers. Range: The minimum point on the graph of this equation is (.5,.5) The range is { :.5} Note: Use nd CALC 3:minimum on the calculator. It is a function because the vertical line test works, not one-to-one. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 09

18 3.3 EffEcts of transformations on ParEnt function GraPHs Parent identit (Linear) Quadratic Eponential square root absolute Value function f() = f() = f() = b ; b > 0, b f() =, 0 f() = f() = m + b (this eample, b = ) Parent Graph f() + a Graph moves up a units f() = + f() = + f() = f() = f() = f() a Graph moves down a units f() = f() = - f() = f() = - - f() = Algebra Made Eas Common Core Edition 8 Copright 0 Topical Review Book Compan Inc. All rights reserved.

19 3. EXPONENTIAL FUNCTIONS FUNCTIONS WITH EXPONENTS Eponential Function: A function with a variable in the eponent. The base of the eponent must be a positive number and not equal to. It is in the form: = b : b > 0, b. The domain of an eponential function is all the real numbers. The range is > 0. Evaluate an Eponential Epression: If given the value of an eponent, appl it to the appropriate base and use the calculator to evaluate. Eamples Given: If =, evaluate each of the following: = = 5 e = e 8 = (e is on the calculator) 8 = 8 = 3 Solving Equations that contain a variable with a constant eponent: Remember that when raising a power to a power, the eponents are multiplied. This rule is used to solve equations where is the base and it is raised to a constant eponent. (See Unit..) Steps: ) Isolate the variable with its eponent. ) Raise both sides of the equation to a power that is equal to the reciprocal of the eponent. This will make the eponent on the variable =. 3) Solve the remaining equation ) Check the answers. Eamples + = 0 Check : = 9 ( 8) + = 0 ( ) = ( 9) 9+ = 0 = 8 0 = 0 Building and Interpreting Functions Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 7

20 SEQUENCES Finding a Specific Term in a Sequence Recursive Definition or Formula: In a recursive definition or formula, the first term in a sequence is given and subsequent terms are defined b the terms before it. If a n is the term we are looking for, a n is the term before it. To find a specific term, terms prior to it must be found. Eample Find the first 3 terms in the sequence a n = 3a n + if a = 5. In this eample, the first term is a = 5. To find the nd and 3 rd terms, n =, and n = 3 need to be substituted. a = 5 a = 3(a ) + ; a = 3(5) + ; a = 9 a 3 = 3(a ) + ; a 3 = 3(9) + ; a 3 = The three terms are 5, 9, and. To write a recursive definition or formula when given several terms in the sequence, it is necessar to find an epression that is developed b comparing the terms and finding the process required to change each term to the subsequent term. Eample Write a recursive definition for this sequence.,,, 5, a =. Since = ( ), and =, and using the last term given to us, 5 =, a recursive definition for this sequence could be a n = (a n ). Eplicit Formula: If specific terms are not given, a formula, sometimes called an eplicit formula, is given. It can be used b substituting the number of the term desired into the formula for n. Simplif as usual. Eamples 3. What is the 7 th term in the sequence a n = n Since we want the 7 th term, n = 7. Substitute 7 in place of n in the equation. a 7 = (7) The 7 th term in this sequence is 0. a 7 = 0 Building and Interpreting Functions What is the 5 th term of the sequence a n = 3 n? Substitute 5 for n. a 5 = 3 5 The 5 th term in this sequence is 3. a 5 = 3 What are the first 3 terms in the sequence a n = n +? 3 calculations are needed: n =, n =, and n = 3. a = + = a = + = 5 The first 3 terms are:, 5, 0 a 3 = 3 + = 0 Note: Terms should be in simplified whenever possible. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 53

21 ARITHMETIC SEQUENCE Each term in the sequence has a common difference, d, with the term preceding it. The first term is labeled a. The formula for finding specific terms of an arithmetic sequence is a n = a + (n )d, where a n is the term desired, a is the first term in the sequence, n is the location in the sequence of the term desired, and d is the common difference. To find d, COMMON DIFFERENCE: a a, a 3 a, etc. If given the first term and the value of d, the formula can be used to find other terms in the sequence. Eamples 3. Find the 7 th term of an arithmetic sequence if a = 5 and d =. a n = a + (n )d a 7 = 5 + (7 )() a 7 = 5 + a 7 = 7 In an arithmetic series a = 5. Find a 0 if a = 7 and a 7 = 9. Find d: a 7 a = 9 7; d = Use formula: a n = a + (n )d a 0 = 5 + (9)() a 0 = 3 Recursive Formula: Terms in the sequence are given to establish a pattern. The general recursive formula for an arithmetic sequence is a n = a n + d but we have to find d. Eample Write the formula for this sequence. {,, 7, 0, 3 } There is a common difference, d, of 3 between each pair of consecutive terms in the sequence. Each term in the sequence is found b adding 3 to the previous term. Since the terms were given, a recursive formula can be developed. a = 3. a n = a n + 3. Find the th term of this sequence: a = a 5 + 3; a = =. The th term of this sequence is. To find the 0 th term of this sequence we would need the 9 th term to use this formula. It would make more sense to use the eplicit formula, a n = a + (n )d. a 0 = a + (0 ) (d); a 0 = + 9(3); a 0 = 58 Algebra Made Eas Common Core Edition 5 Copright 0 Topical Review Book Compan Inc. All rights reserved.

22 3. GEOMETRIC SEQUENCE The consecutive terms are developed b multipling each term in the sequence b a common ratio, r, to obtain the net consecutive term. The terms in the a sequence have a common ratio, r =. (Some tets refer to a geometric a sequence as a geometric progression.) To find r, the COMMON RATIO: Divide a term b the term before a a it. r = ; r = An two consecutive terms in a geometric a a3 sequence will have the same common ratio. Eamples 3,,,... Each pair of terms a a a 3 a has a ratio of. 3 =, =, =. Each term in this sequence is found b multipling the previous term b , 9, 3... r= =, r= = CommonRatio r = a 8 a a 3 Each term in this sequence is multiplied b /3 to get the net term. Finding a Specific Term of a Geometric Sequence: Use the formula a n = a r n where n is the inde of the term desired, r is the common ratio of the sequence and a is the first term of the sequence. Determine the value of r first, then substitute and simplif. Using the sequence in eample above, 3,,,, find the th term n =, r =, a = 3 a = a r n a = (3)( ( ) ) a = 3(08) a = Find the 7 th term in this sequence:,, 8 a 7 ; a ; a = ( ) = ( ) ( ) = First find r : r = = 8 The Recursive Formula for a geometric sequence is a n = (a n )r when terms are given. Eample What is the 5 th term of the sequence 5, 0, 0, 0, r = 0 =, 0 = ; r=, a n - = 0 0 is the 0 0 th term a 5 = (0)() = 80 Building and Interpreting Functions Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 55

23 Unit PROBABILITY & STATISTICS Make inferences and justif conclusions from sample surves, eperiments, and observational studies. Understand independence and conditional probabilit and use them to interpret data. Use the rules of probabilit to compute probabilities of compound events in a uniform probabilit model. Summarize, represent, and interpret data on a single count or measurement variable. Understand and evaluated random processes underling statistical eperiments. Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 59

24 . Statistics is a process for collecting and analzing data in large quantities, especiall for the purpose of inferring population characteristics based on a random sample from that population. Kinds of Data Studies: The data can be collected in a variet of was. These include: Population vs Sample The tpe and number of people who participate in a statistical stud can impact the validit of the stud. Data collection that includes all members of a population, is called a census. If onl part of a population is in the stud it is called a sample. In a sample, the data can be epanded to include the whole group based on the epectation that the information gathered would appl to the group as a whole. We use samples to make conclusions about the entire population. The problem with samples is that the ma not trul represent the population. A sample is considered random if the probabilit of selecting the sample is the same as the probabilit of selecting ever other sample. When a sample is not random, a bias is introduced which ma influence the stud in favor of one outcome over other outcomes. A good sample must:. represent the whole population.. be large enough. 3. be randoml selected to eliminate bias. Surve Used to gather large quantities of facts or opinions. Eample Political parties call to ask people to name their favorite candidate. Observation The observer does not have an interaction with the subjects and just eamines the results of an activit. Eample Count the number of people who are using a cell phone in a mall in a particular time frame. Count the number of people wearing sneakers at school. Controlled Eperiment Two groups are studied while an eperiment is performed with one of them but not the other. Eample The value of drinking orange juice to prevent a cold is measured b seeing how man people in a group given orange juice to drink for a month get a cold vs how man get a cold in the group that did not drink orange juice. Algebra Made Eas Common Core Edition 0 Copright 0 Topical Review Book Compan Inc. All rights reserved. STATISTICS

25 . TYPES OF VARIABLES AND DATA Categorical Variable: Allows the identification of the group or categor in which the individual is placed. Quantitative Variable: Results are numerical which allows arithmetic to be performed on them. Categorical Data are non-numerical. The data values are identified b tpe. It is also called qualitative data. Eample Ee color, kind of pet owned, or the name of a favorite TV show or movie. Quantitative Data are numerical. The data values (or items) are measurements or counts and have meaning as numbers. Eample Grades on a test, hours watching TV, or heights of students. Univariate: Measurements are made on onl one variable per observation. Eample Quantitative: Ages of the students in a club. Categorical: Kind of car owned. Bivariate: Measurements that show a relationship between two variables. Eample Quantitative: Grade level and age of the students in a school. Categorical: Gender and favorite TV shows. COLLECTION OF DATA Data collection must be done randoml to obtain reliable results. A sample is often used to infer the characteristics of a population. Interpreting Categorical and Quantitive Data Biased: A data set that is obtained that is likel to be influenced b something giving a slant to the results. Eample Quantitative: To determine the average age of high school students b asking onl tenth graders how old the are. Categorical: Standing outside Yankee Stadium and asking people coming out of the stadium to name their favorite baseball team. Most would sa Yankees! Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved.

26 . Two-Wa Frequenc Tables Tables are often used as a tool to organize and present data. Two-wa frequenc tables are used to organize bivariate categorical data in rows and columns and interpret the data. Two-wa frequenc tables can be used to calculate probabilities. Eamples PROBABILITIES USING TWO-WAY TABLES AND VENN DIAGRAMS The table below shows the results of a surve in which oung adults ages 8- were asked if the ever used Instagram (es or no). Female (F) Male (M) TOTAL Has Used Instagram (Y) Has Never Used Instagram (N) 5 8 TOTAL a) Use the table above to find the probabilit of randoml selecting a oung adult who is female (F) and has used Instagram (Y) to the nearest percent. There are females who have used Instagram out of total of 50 oung adults. The probabilit of a oung adult who is female and has used Instagram is. Therefore, % of the oung adults are female 50 and have used Instagram. b) Use the table above to find the probabilit of randoml selecting a oung adult who is female or has used Instagram to the nearest percent. Eplain wh P(F or Y ) P(F ) + P(Y ). Solution: P(F ) = students are girls. P(Y ) = students use Instagram. P(F and Y ) = 50 girls use Instagram. P(F or Y ) = P(F ) + P(Y ) P(F and Y ) = = = =.8 Conclusion: The probabilit of randoml selecting a oung adult who is female or has used Instagram to the nearest percent is 87%. P(F or Y ) P(F ) + P(Y ) because there was overlapping. P(F ) + P(Y ) = = 5 which is not even a possible probabilit Algebra Made Eas Common Core Edition 98 Copright 0 Topical Review Book Compan Inc. All rights reserved.

27 Algebra Made Eas Handbook Common Core Standards Edition Correlation of Standards Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 0

28 CORRELATIONS TO COMMON CORE STATE STANDARDS Common Core State Standards Unit #. Section # Arithmetic with Polnomials and Rational Epressions (A.APR) A.APR...5 A.APR...5,. Creating Equations (A.CED) A.CED....,. Reasoning with Equations and Inequalities (A.REI) A.REI... A.REI...7,.8,.9 A.REI... A.REI...0 A.REI Interpreting Functions (F.IF) F.IF , 3. F.IF...3., 3.3 F.IF...3. F.IF , 3. F.IF , 3., 3.7, 3.0 Building Functions (F.BF) F.BF , 3. F.BF F.BF , 3. F.BF.a Linear, Quadratic, and Eponential Models (F.LE) F.LE F.LE...3., 3.7, 3.8 F.LE Algebra Made Eas Common Core Edition 0 Copright 0 Topical Review Book Compan Inc. All rights reserved.

29 CORRELATIONS TO COMMON CORE STATE STANDARDS Trigonometric Functions (F.TF) F.TF...,.,.3 F.TF... F.TF.5...,.5,. F.TF Epressing Geometric Properties with Equations (G.GPE) G.GPE... Making Inferences and Justifing Conclusions (S.IC) S.IC...,. S.IC...,.5 S.IC.3...,. S.IC...,.,.8 S.IC.5...,. S.IC...,. Interpreting Categorical and Quantitative Data (S.ID) S.ID...3,. S.ID...5,. Conditional Probabilit and the Rules of Probabilit (S.CP) S.CP...7,.8,.9 S.CP...0 S.CP ,. S.CP....5,.,.9,.0,. S.CP ,. S.CP...0 S.CP Correlations Algebra Made Eas Common Core Edition Copright 0 Topical Review Book Compan Inc. All rights reserved. 03